| L(s) = 1 | − 2·3-s − 2·7-s + 3·9-s + 11-s − 3·13-s − 17-s + 4·21-s − 6·23-s − 4·27-s + 6·29-s + 31-s − 2·33-s + 7·37-s + 6·39-s + 3·41-s − 8·43-s + 3·49-s + 2·51-s − 53-s + 3·59-s + 5·61-s − 6·63-s + 11·67-s + 12·69-s + 11·71-s − 4·73-s − 2·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 9-s + 0.301·11-s − 0.832·13-s − 0.242·17-s + 0.872·21-s − 1.25·23-s − 0.769·27-s + 1.11·29-s + 0.179·31-s − 0.348·33-s + 1.15·37-s + 0.960·39-s + 0.468·41-s − 1.21·43-s + 3/7·49-s + 0.280·51-s − 0.137·53-s + 0.390·59-s + 0.640·61-s − 0.755·63-s + 1.34·67-s + 1.44·69-s + 1.30·71-s − 0.468·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.456504525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456504525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 68 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 66 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 29 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + T + 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 102 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 146 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 154 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 23 T + 280 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 186 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.432900534059524565307309911669, −8.259102536748075376465699561127, −7.81352686197080408486055807811, −7.36861147442174957557065926731, −7.07550235424287441480150289425, −6.53783865249907647766328826451, −6.48368080330438695087703294226, −6.11575469534286484635155993589, −5.57917755822608596064841158640, −5.40168893895487266629382908293, −4.76155253480805716097704850496, −4.60641369821744260838514393309, −3.96168328780885649427828716647, −3.90023369393842844626705197283, −2.96371702022899953012935617012, −2.90788068191557324368974869451, −1.96518553311349354715269693858, −1.86284926821227334835172016217, −0.68981988804449127181538191248, −0.58293113543591519769079203080,
0.58293113543591519769079203080, 0.68981988804449127181538191248, 1.86284926821227334835172016217, 1.96518553311349354715269693858, 2.90788068191557324368974869451, 2.96371702022899953012935617012, 3.90023369393842844626705197283, 3.96168328780885649427828716647, 4.60641369821744260838514393309, 4.76155253480805716097704850496, 5.40168893895487266629382908293, 5.57917755822608596064841158640, 6.11575469534286484635155993589, 6.48368080330438695087703294226, 6.53783865249907647766328826451, 7.07550235424287441480150289425, 7.36861147442174957557065926731, 7.81352686197080408486055807811, 8.259102536748075376465699561127, 8.432900534059524565307309911669
